Design and Analysis of Algorithm

Design and Analysis of Algorithm

SNSCT – Department of Compute Science and Engineering Page 1

SNS COLLEGE OF TECHNOLOGY

COIMBATORE – 35

DEPARTMENT OF COMPUTER SIENCE AND ENGINEERING (UG & PG)

Second Year Computer Science and Engineering, 4rd Semester

2 Marks Question and Answer

Subject Code & Name: Design and Analysis of Algorithm

Prepared by: J. kasthuri L/CSE, S.P Siddque Ibrahim L/CSE

UNIT – I

1. What is performance measurement?

Performance measurement is concerned with obtaining the space and the

time

requirements of a particular algorithm.

2. What is an algorithm?

An algorithm is a finite set of instructions that, if followed, accomplishes a

particular task.

3. What are the characteristics of an algorithm?

1) Input

2) Output

3) Definiteness

4) Finiteness

5) Effectiveness

4. Define Program.

A program is the expression of an algorithm in a programming language.

Sometimes works such as procedure, function and subroutine are used

synonymously

program.

5. Write the For LOOP general format.

The general form of a for Loop is

For variable : = value 1 to value 2 step

Step do

{

<statement 1>

<statement n >

}



6. What is recursive algorithm?

An algorithm is said to be recursive if the same algorithm is invoked in the

body. An algorithm that calls itself is Direct recursive. Algorithm A is said

to be

indeed recursive if it calls another algorithm, which in turn calls A.

7. What is space complexity?

The space complexity of an algorithm is the amount of memory it needs

to run

to completion.

8. What is time complexity?

The time complexity of an algorithm is the amount of computer time it

needs

to run to completion.

9. Give the two major phases of performance evaluation

Performance evaluation can be loosely divided into two major phases:

(i) a prior estimates (performance analysis)

(ii) a Posterior testing(performance measurement)

10. Define input size.

The input size of any instance of a problem is defined to be the number of

words(or the number of elements) needed to describe that instance.

11. Define best-case step count.

The best-case step count is the minimum number of steps that can be

executed

for the given parameters.

12. Define worst-case step count.

The worst-case step count is the maximum number of steps that can be

executed for the given parameters.

13. Define average step count.

The average step count is the average number of steps executed an

instances

with the given parameters.

14. Define the asymptotic notation “Big on” (0)

The function f(n) = O(g(n)) iff there exist positive constants C and no such

that f(n)£ C * g(n) for all n, n ³n0.



15. Define the asymptotic notation “Omega” ( W ).

The function f(n) =W (g(n)) iff there exist positive constant C and no such

that

f(n) C * g(n) for all n, n ³ n0.

16. Define the asymptotic t\notation “theta” (q )

The function f(n) =q (g(n)) iff there exist positive constant C1, C2, and no

such that C1 g(n)£ f(n) £ C2 g(n) for all n, n ³ n0.

17. Define Little “oh”.

The function f(n) = 0(g(n))

Iff Lim f(n) = 0

n - μ g(n)

18. Define Little Omega.

The function f(n) = w (g(n))

iff

Lim f(n) = 0

n - μ g(n)

19. Write algorithm using iterative function to fine sum of n

numbers.

Algorithm sum(a,n)

{

S : = 0.0

For i=1 to n do

S : - S + a[i];

Return S;

}

20. Write an algorithm using Recursive function to fine sum of n

numbers.

Algorithm Rsum (a,n)

{

If(n_ 0) then

Return 0.0;

Else Return Rsum(a, n- 1) + a(n);

}

21. What are the basic asymptotic efficiency classes?

The various basic efficiency classes are

Constant:1

Logarithmic: log n



Linear: n

N-log-n: n log n

Quadratic: n2

Cubic: n3

Exponential:2n

Factorial:n!

22. Define Time Space Tradeoff.

Time and space complexity can be reduced only to certain levels, as later

on

reduction of time increases the space and vice-versa. This is known as

Time-space

trade-off.

23. Why do we need algorithm analysis?

Writing a working program is not good enough.

The program may be in-efficient

If the program is run on a large data set, then the running time becomes

an

issue.

24. List the factors which affects the running time of the

algorithm.

A. Computer

B. Compiler

C. Algorithm used

D. Input to the algorithm

i. The content of the input affects the running time

ii. Typically, the input size is the main consideration.

25. What is recurrence equation?

A recurrence equation is an equation or inequality that describes a

function in

terms of its value on smaller inputs.

26. What is classification of the recurrence equation?

Recurrence Equations can be classified into

1. Homogeneous

2. Inhomogeneous



12 mark Questions

Unit I

1. Define the asymptotic notations used for best case average case and

worst case

analysis of algorithms.

2. Write an algorithm for finding maximum element of an array , perform

best , worst

and average case complexity with appropriate order notations.

3. Write an algorithm to find mean and variance of an array perform best,

worst and

average

4. Explain the various criteria used for analyzing algorithms.

5. List the properties of various asymptotic notations.

6. Explain the necessary steps for analyzing the efficiency of recursive

algorithms.

7. Write short notes on algorithm visualization.

8. Describe briefly the notions of complexity of an algorithm.

9. What is pseudo-code? Explain with an example

10. Find the complexity (C (n))of the algorithm for the worst case, best

case and average

case. (evaluate average case complexity for n = 3,where n is number of

inputs)



UNIT II

1. Define the divide and conquer method.

Given a function to compute on ‘n’ inputs the divide-and-comquer

strategy

suggests splitting the inputs in to’k’ distinct susbsets, 1<k <n, yielding ‘k’

subproblems. The subproblems must be solved, and then a method must

be found

tocombine subsolutions into a solution of the whole. If the subproblems

are still

relatively large, then the divide-and conquer strategy can possibly be

reapplied.

2. Define control abstraction.

A control abstraction we mean a procedure whose flow of control is clear

but

whose primary operations are by other procedures whose precise

meanings are left

undefined.

3. Write the Control abstraction for Divide-and conquer.

Algorithm D And(r)

{

if small(p) then return S(r);

else

{

divide P into smaller instance _ 1, _ 2, _ k, k ³ 1;

Apply D and C to each of these subproblems

Return combine (DAnd C(_1) DAnd C(_2),----, DAnd ((_k));

}

}

4. What is the substitution method?

One of the methods for solving any such recurrence relation is called the

substitution method.

5. What is the binary search?

If ‘q’ is always chosen such that ‘aq’ is the middle element(that is,

q=[(n+1)/2), then the resulting search algorithm is known as binary

search.

6. Give computing time for Bianry search?

The computing time of binary search by giving formulas that describe the

best,

average and worst cases. Successful searches q(1) q(logn) q(Logn) best

average worst

unsuccessful searches q(logn) best, average, worst



7. Define external path length?

The external path length E, is defines analogously as sum of the distance

of all

external nodes from the root.

8. Define internal path length.

The internal path length ‘I’ is the sum of the distances of all internal nodes

from the root.

9. What is the maximum and minimum problem?

The problem is to find the maximum and minimum items in a set of ‘n’

elements.Though this problem may look so simple as to be contrived, it

allows us to

demonstrate divide and conquer in simple setting.

10. What is the Quick sort?

n quicksort, the division into subarrays is made so that the sorted sub

arrays do

not need to be merged later.

11. Write the Anlysis for the Quick sot.

In analyzing QUICKSORT, we can only make the number of element

comparisions c(n). It is easy to see that the frequency count of other

operations is of

the same order as C(n).

12. Is insertion sort better than the merge sort?

Insertion sort works exceedingly fast on arrays of less then 16 elements,

though for large ‘n’ its computing time is O(n2).

13. Write a algorithm for straightforward maximum and

minimum>

algorithm straight MaxMin(a,n,max,min)

//set max to the maximum and min to the minimum of a[1:n]

{

max := min: = a[i];

for i = 2 to n do

{

if(a[i] >max) then max: = a[i];

if(a[i] >min) then min: = a[i];

}

}

14. Give the recurrence relation of divide-and-conquer?



The recurrence relation is

T(n) = g(n)

T(n1) + T(n2) + ----+ T(nk) + f(n)

15. Write the algorithm for Iterative binary search?

Algorithm BinSearch(a,n,x)

//Given an array a[1:n] of elements in nondecreasing

// order, n>0, determine whether x is present

{

low : = 1;

high : = n;

while (low < high) do

{

mid : = [(low+high)/2];

if(x < a[mid]) then high:= mid-1;

else if (x >a[mid]) then low:=mid + 1;

else return mid;

}

return 0;

}

16. What are internal nodes?

The circular node is called the internal nodes.

17. Describe the recurrence relation ofr merge sort?

If the time for the merging operation is proportional to n, then the

computing

time of merge sort is described by the recurrence relation

n = 1, a a constant

T(n) = a

2T (n/2) + n n >1, c a constant

18. What is meant by feasible solution?

Given n inputs and we are required to form a subset such that it satisfies

some

given constraints then such a subset is called feasible solution.

19. Write any two characteristics of Greedy Algorithm?

o To solve a problem in an optimal way construct the solution from given

set of

candidates.

o As the algorithm proceeds, two other sets get accumulated among this

one set

contains the candidates that have been already considered and chosen

while

the other set contains the candidates that have been considered but

rejected.

20. Define optimal solution?



A feasible solution either maximizes or minimizes the given objective

function

is called as optimal solution

21. What is Knapsack problem?

A bag or sack is given capacity n and n objects are given. Each object has

weight wi and profit pi .Fraction of object is considered as xi (i.e)

0<=xi<=1 .If

fraction is 1 then entire object is put into sack. When we place this

fraction into the

sack we get wixi and pixi.

22. Define weighted tree.

A directed binary tree for which each edge is labeled with a real number

(weight) is called as weighted tree.

23. What is the use of TVSP?

In places where the loss exceeds the tolerance level boosters have to the

placed. Given a network and loss tolerance level the tree vertex splitting

problems is

to determine an optimal placement of boosters.

24. What is the Greedy choice property?

The first component is greedy choice property (i.e.) a globally optimal

solution can arrive at by making a locally optimal choice.

The choice made by greedy algorithm depends on choices made so far

but it

cannot depend on any future choices or on solution to the sub problem.

It progresses in top down fashion.

25. What is greedy method?

Greedy method is the most important design technique, which makes a

choice

that looks best at that moment. A given ‘n’ inputs are required us to

obtain a subset

that satisfies some constraints that is the feasible solution. A greedy

method suggests

that one can device an algorithm that works in stages considering one

input at a time.

26. What are the steps required to develop a greedy algorithm?

Determine the optimal substructure of the problem.

Develop a recursive solution.

Prove that at any stage of recursion one of the optimal choices is greedy

choice.

Thus it is always safe to make greedy choice.

Show that all but one of the sub problems induced by having made the

greedy

choice are empty.

Develop a recursive algorithm and convert into iterative algorithm.



27. What is activity selection problem?

The ‘n’ task will be given with starting time si and finishing time fi.

Feasible

Solution is that the task should not overlap and optimal solution is that

the task should

be completed in minimum number of machine set.

28. Write the specification of TVSP

Let T= (V, E, W) be a weighted directed binary tree where

V_ vertex set

E_ edge set

W_ weight function for the edge.

W is more commonly denoted as w (i,j) which is the weight of the edge

<i,j> _

E.

29. Define forest.

Collection of sub trees that are obtained when root node is eliminated is

known as forest

30. What does Job sequencing with deadlines mean?

Given a set of ‘n’ jobs each job ‘i’ has a deadline di such that di>=0 and

a

profit pi such that pi>=0.

For job ‘i’ profit pi is earned iff it is completed within deadline.

Processing the job on the machine is for 1unit of time. Only one machine

is

available.

31. Define post order traversal.

The order in which the TVSP visits the nodes of the tree is called the post

order traversal.

32. Write the formula to calculate delay and also write the

condition in which the

node gets splitted?

To calculate delay

d(u)=max{d(v)+w(u, v)}

v _ c(u)

The condition in which the node gets splitted are:

d(u)+w(u ,v)>_ and also d(u) is set to zero.

33. What is meant by tolerance level?

The network can tolerate the losses only up to a certain limit tolerance

limit.



34. When is a task said to be late in a schedule?

A task is said to be late in any schedule if it finishes after its deadline else

a

task is early in a schedule.

35. Define feasible solution for TVSP.

Given a weighted tree T(V,E,W) and a tolerance limit _ any subset X of V is

a

feasible solution if d(T/X).

36. Define optimal solution for TVSP.

An optimal solution is one in which the number of nodes in X is minimized.

37. Write the general algorithm for Greedy method control

abstraction.

Algorithm Greedy (a, n)

{

solution=0;

for i=1 to n do

{

x= select(a);

if feasible(solution ,x) then

solution=Union(solution ,x);

}

return solution;

}

38. Define optimal solution for Job sequencing with deadlines.

Feasible solution with maximum profit is optimal solution for Job

sequencing

with deadlines.

39. Write the difference between the Greedy method and

Dynamic programming.

Greedy method Dynamic programming

Greedy method Dynamic Programming

_ Only one sequence of decision is Many number of decisions are

generated. generated1.

_ It does not guarantee to give an It definitely gives an optimal

solution always. optimal solution always.



12 mark Questions

Unit II

1. Sort the following set of elements using Quick Sort. (24,8,71, 4,23,6)

2. Give a detailed note on divide and conquer techniques.

3. Write an algorithm for searching an element using binary search

Method, Give an

example.

4. Compare and contrast BFS and DFS.

5. Explain the merge sort.

6. Explain the method of finding the minimum spanning tree for a

connected graph using

Prim's algorithm.

7. Discuss the 0/1 knapsack problem.



UNIT III

1. Define dynamic programming.

Dynamic programming is an algorithm design method that can be used

when a

solution to the problem is viewed as the result of sequence of decisions.

2. What are the features of dynamic programming?

Optimal solutions to sub problems are retained so as to avoid

recomputing

their values.

Decision sequences containing subsequences that are sub optimal are

not

considered.

It definitely gives the optimal solution always.

3. What are the drawbacks of dynamic programming?

Time and space requirements are high, since storage is needed for all

level.

Optimality should be checked at all levels.

4. Write the general procedure of dynamic programming.

The development of dynamic programming algorithm can be broken into

a

sequence of 4 steps.

1. Characterize the structure of an optimal solution.

2. Recursively define the value of the optimal solution.

3. Compute the value of an optimal solution in the bottom-up fashion.

4. Construct an optimal solution from the computed information.

5. Define principle of optimality.

It states that an optimal sequence of decisions has the property that

whenever

the initial stage or decisions must constitute an optimal sequence with

regard to

stage resulting from the first decision.

6. Give an example of dynamic programming and explain.

An example of dynamic programming is knapsack problem. The solution

to

the knapsack problem can be viewed as a result of sequence of decisions.

We have

to decide the value of xi for 1<i<n. First we make a decision on x1 and

then on x2

and so on. An optimal sequence of decisions maximizes the object

function Spi xi.



7. Write about optimal merge pattern problem.

Two files x1 and x2 containing m & n records could be merged together to

obtain one merged file. When more than 2 files are to be merged

together. The merge

can be accomplished by repeatedly merging the files in pairs. An optimal

merge

pattern tells which pair of files should be merged at each step. An optimal

sequence

is a least cost sequence.

8. Explain any one method of finding the shortest path.

One way of finding a shortest path from vertex i to j in a directed graph G

is to

decide which vertex should be the second, which is the third, which is the

fourth, and

so on, until vertex j is reached. An optimal sequence of decisions is one

that results

in a path of least length.

9. Define 0/1 knapsack problem.

The solution to the knapsack problem can be viewed as a result of

sequence of

decisions. We have to decide the value of xi. xi is restricted to have the

value 0 or 1

and by using the function knap(l, j, y) we can represent the problem as

maximum Spi

xi subject to Swi xi < y where l -iteration, j - number of objects, y –

capacity.

10. What is the formula to calculate optimal solution in 0/1

knapsack problem?

The formula to calculate optimal solution is

g0(m)=max{g1, g1(m-w1)+p1}.

11. Write about traveling salesperson problem.

Let g = (V, E) be a directed. The tour of G is a directed simple cycle that

includes every vertex in V. The cost of a tour is the sum of the cost of the

edges on

the tour.The traveling salesperson problem to find a tour of minimum

cost.

12. Write some applications of traveling salesperson problem.

Routing a postal van to pick up mail from boxes located at n different

sites.

Using a robot arm to tighten the nuts on some piece of machinery on an

assembly line.

Production environment in which several commodities are manufactured

on

the same set of machines.

13. Give the time complexity and space complexity of traveling

salesperson

problem.



Time complexity is O (n2 2n).

Space complexity is O (n 2n).

14. Define flow shop scheduling.

The processing of jobs requires the performance of several distinct job. In

flow shop we have n jobs each requiring n tasks i.e. T1i, T2i,…...Tmi,

1<i<n.

15. What are the conditions of flow shop scheduling?

Let Tji denote jth task of the ith job. Task Tij is to be performed on Pj

number of

processors where 1<j<m i.e. number of processors will be equal to

number of

task

Any task Tji must be assigned to the processor Pj.

No processor can have more than one task assigned to it at any time.

For any jobI

processing the task for j>1 cannot be started until T(j-i),i has been

completed.

16. Define nonpreemptive schedule.

A non preemptive schedule is a schedule in which the processing of a task

on

any processor is not terminated until the task is complete.

17. Define preemptive schedule.

A preemptive schedule is a schedule in which the processing of a task on

any

processorcan be terminated before the task is completed.

18. Define finish time

The finish time fi (S) of job i is the time at which all tasks of job i have

been

completed in schedule S.The finish time F(S) of schedule S is given by

F(S)=max{ fi

(S)} 1<i<n

19. Define mean flow time

The mean flow time MFT (S) is defined to be]

MFT (S) = 1 Sfi(S)

n 1<i<n

20. Define optimal finish time.

Optimal finish time scheduling for a given set of tasks is a nonpreemptive

schedule S for which F (S) is minimum over all nonpreemptive schedules

S.

21. Define preemptive optimal finish time.

Preemptive optimal finish time scheduling for a given set of tasks is a

preemptive schedule S for which F (S) is minimum over all preemptive

schedules S.



22. What are the requirements that are needed for performing

Backtracking?

To solve any problem using backtracking, it requires that all the solutions

satisfy a complex set of constraints. They are:

i. Explicit constraints.

ii. Implicit constraints.

23. Define explicit constraint.

They are rules that restrict each xi to take on values only from a give set.

They

depend on the particular instance I of the problem being solved. All tuples

that

satisfy the explicit constraints define a possible solution space.

24. Define implicit constraint.

They are rules that determine which of the tuples in the solution space of I

satisfy the criteria function. It describes the way in which the xi must

relate to

each other.

25. Define state space tree.

The tree organization of the solution space is referred to as state space

tree.

26. Define state space of the problem.

All the paths from the root of the organization tree to all the nodes is

called as

state space of the problem

27. Define answer states.

Answer states are those solution states s for which the path from the root

to s

defines a tuple that is a member of the set of solutions of the problem.

28. What are static trees?

The tree organizations that are independent of the problem instance

being

solved are called as static tree.

29. What are dynamic trees?

The tree organizations those are independent of the problem instance

being

solved are called as static tree.

30. Define a live node.



A node which has been generated and all of whose children have not yet

been

generated is called as a live node.

31. Define a E – node.

E – node (or) node being expanded. Any live node whose children are

currently being generated is called as a E – node.

32. Define a dead node.

Dead node is defined as a generated node, which is to be expanded

further all

of whose children have been generated.

Unit III

1 Describe the travelling salesman problem and discuss how to solve it

using dynamic

programming.

2 Discuss the use of Greedy method in solving Knapsack problem and

subset-sum

problem.

3 Write an algorithm to sort a set of " M "numbers using insertion sort.

4 How will you find the shortest path between two given vertices using

Dijkstra's

algorithm? Explain.

5 Explain multistage graphs and give it example.





UNIT IV

1.,What are the factors that influence the efficiency of the

backtracking algorithm?

The efficiency of the backtracking algorithm depends on the following four

factors. They are:

i. The time needed to generate the next xk

ii. The number of xk satisfying the explicit constraints.

iii. The time for the bounding functions Bk

iv. The number of xk satisfying the Bk.

2.Define Branch-and-Bound method.

The term Branch-and-Bound refers to all the state space methods in which

all

children of the E-node are generated before any other live node can

become the E- node.

3.What are the searching techniques that are commonly used in

Branch-and-Bound

method.

The searching techniques that are commonly used in Branch-and-Bound

method

are:

i. FIFO

ii. LIFO

iii. LC

iv. Heuristic search

4.State 8 – Queens problem.

The problem is to place eight queens on a 8 x 8 chessboard so that no two

queen

“attack” that is, so that no two of them are on the same row, column or

on the diagonal.

5.State Sum of Subsets problem.

Given n distinct positive numbers usually called as weights , the problem

calls for finding all

the combinations of these numbers whose sums are m.

6. State m – colorability decision problem.

Let G be a graph and m be a given positive integer. We want to discover

whether the nodes of

G can be colored in such a way that no two adjacent nodes have the same

color yet only m

colors are used.

7.Define chromatic number of the graph.

The m – colorability optimization problem asks for the smallest integer m

for which the

graph G can be colored. This integer is referred to as the chromatic

number of the graph.

8. Define a planar graph.

A graph is said to be planar iff it can be drawn in such a way that no two

edges cross each

other.

9. What are NP- hard and Np-complete problems?

The problems whose solutions have computing times are bounded by

polynomials of small

degree.

10. What is a decision problem?

Any problem for which the answer is either zero or one is called decision

problem.



11. What is maxclique problem?

A maxclique problem is the optimization problrm that has to determine

the size of a largest

clique in Grapg G where clique is the maximal subgraph of a graph.

12. what is approximate solution?

A feasible solution with value close to the value of an optimal solution is

called

approximate solution.

Design and Analysis of Algorithm

Documents

n gn

recursive algorithm

function fn

particular algorithm

step step

c1 gn fn c2 gn

w gn iff lim fn

q gn iff